Energy Transfers and Stopping Power

In CasP fast numerical calculations of the mean electronic energy transfer
Q_e (due to excitation and ionization of target atoms) are performed
for each individual impact parameterb
in a collision. The total electronic energy-loss
cross-section S_e (equivalent to the stopping power) is subsequently
calculated from Q_e(b). The computation of Q_e and S_e
accounts for a selected predefined projectile-screening
function. By selecting a proper screening function, it is possible to treat
non-equilibrium energy-loss phenomena. Furthermore, the unitary convolution
approximation UCA (default selection)
is a non-linear theory, as it includes the Bloch terms. Therefore, even
for bare projectiles, the results do not scale with the square of the projectile
charge (as most other quantum theories do).

The so-called electron-capture and loss cycles may be important for energies
close to or below the matching velocity, where the projectile speed is equal
to the orbital velocity of the target-electron shell under consideration (roughly
25 keV/u for the valence bands). At higher projectile velocities the energy
transfer to target electrons becomes by far the dominant contribution to the
slowing down of ions. For most screening options, only this electronic energy
transfer to neutral target atoms - mechanism (a) - is included in the current
version of the program CasP. A specific screening option, however, allows to
calculate the contribution to equilibrium stopping powers due to projectile-electron
loss plus electron capture.

The program CasP makes use of the convolution
approximation (either the perturbative convolution approximation PCA
or the more advanced unitary convolution approximation UCA).
The physical inputs of the program are the
projectile velocity, the projectile-screening potential, the target-electron
density distribution (which we have tabulated using results from our Hartree-Fock-Slater
code) and the oscillator strengths for the
target electrons. The code is based on an exact matching of the quantum mechanical
mean electronic energy transfers for the asymptotic region of very low and very
high energy transfers to the target electron, similar as in the Bethe or in
the Bloch theory.

It includes a simplified impact-parameter dependent relativistic correction
(no radiation energy loss and no relativistic density effect) and thus,
it becomes inaccurate for kinetic energies per nucleon exceeding
several 100 MeV/u.

The PCA is an impact-parameter dependent approximation to the first
Born approximation (1st order perturbation theory, often also denoted
semi-classical approximation SCA or, for total cross sections, Bethe theory).
It is based the assumption of a classical straight-line motion of the projectile,
which is well justified as long as the projectile is faster than a few keV/u
(if projectile scattering angles are detected or energy loss due to inner-shell
electrons is considered, the range of validity of this approximation shifts
to higher energies). The PCA approximation converges slightly more rapidly towards
the exact 1st order results for the total stopping cross section
than the simple Bethe formula.

The UCA includes an approximation to the wavepacket formalism
by Bloch, and thus each electron is counted only once in a collision
and ionization probabilities are restricted to a maximum of 100%
(unitarity). This is not the case for the PCA, which correspondingly is
less suited for the prediction of heavy-ion stopping powers. Note, that
the current version of the UCA should be accurate for very light and
very heavy ions. Futhermore, polarization effects (e.g., the Barkas
effect) due to close collisions are now included in the current
version of CasP.